Question

Write an exponential function that models the points in the table remember your function should be write y-a(b)^x
X:0,1,2,3
Y:-3,-9,-27,-81

Answer 1

To model the points in the table with an exponential function of the form \( y = a(b)^x \), we first need to identify a and b using the given values of x and y.

From the table:
- When \( x = 0 \), \( y = -3 \)
- When \( x = 1 \), \( y = -9 \)
- When \( x = 2 \), \( y = -27 \)
- When \( x = 3 \), \( y = -81 \)

### Step 1: Find 'a'.

Using the first point \( (0, -3) \):
\[
y = a(b)^0 \implies y = a \implies -3 = a
\]
So, \( a = -3 \).

### Step 2: Find 'b'.

Now we can use the second point \( (1, -9) \) to find b:
\[
y = -3(b)^1 \implies -9 = -3b
\]
Dividing both sides by \(-3\):
\[
b = \frac{-9}{-3} = 3
\]

### Step 3: Verify with other points.

To ensure this is consistent, we can check other points:

1. For \( x = 2 \):
\[
y = -3(3)^2 = -3(9) = -27 \quad \text{(Correct)}
\]

2. For \( x = 3 \):
\[
y = -3(3)^3 = -3(27) = -81 \quad \text{(Correct)}
\]

### Conclusion

The exponential function that models the points in the table is:
\[
y = -3(3)^x
\]